The dynamical attainability of ESS in evolutionary games

In this paper, the attainability of ESS of the evolutionary game among n players under the frequency-independent selection is studied by means of a mathematical model describing the dynamical development and a concept of stability (strongly determined stability). It is assumed that natural selection and small mutations cause the phenotype to change gradually in the direction of fitness increasing. It is shown that (1) the ESS solution is not always evolutionarily attainable in the evolutionary dynamics, (2) in the game where the interaction between two species is completely competitive, the Nash solution is always attainable, and (3) one of two species may attain the state of minimum fitness as a result of evolution. The attainability of ESS is also examined in two game models on the sex ratio of wasps and aphids in light of our criterion of the attainability of ESS.     

A predator–prey refuge system: Evolutionary stability in ecological systems

A refuge model is developed for a single predator species and either one or two prey species where no predators are present in the prey refuge. An individual’s fitness depends on its strategy choice or ecotype (predators decide which prey species to pursue and prey decide what proportion of their time to spend in the refuge) as well as on the population sizes of all three species. It is shown that, when there is a single prey species with a refuge or two prey species with no refuge compete only indirectly (i.e. there is only apparent competition between prey species), that stable resident systems where all individuals in each species have the same ecotype cannot be destabilized by the introduction of mutant ecotypes that are initially selectively neutral. In game-theoretic terms, this means that stable monomorphic resident systems, with ecotypes given by a Nash equilibrium, are both ecologically and evolutionarily stable. However, we show that this is no longer the case when the two indirectly-competing prey species have a refuge. This illustrates theoretically that two ecological factors, that are separately stabilizing (apparent competition and refuge use), may have a combined destabilizing effect from the evolutionary perspective. These results generalize the concept of an evolutionarily stable strategy (ESS) to models in evolutionary ecology. Several biological examples of predator–prey systems are discussed from this perspective.     

Evolutionary stability for large populations

We present a revision of Maynard Smith's evolutionary stability criteria for populations which are very large (though technically finite) and of unknown size. We call this the large population ESS, as distinct from Maynard Smith's infinite population ESS and Schaffer's finite population ESS. Building on Schaffer's finite population model, we define the large population ESS as a strategy which cannot be invaded by any finite number of mutants, as long as the population size is sufficiently large. The large population ESS is not equivalent to the infinite population ESS: we give examples of games in which a large population ESS exists but an infinite population ESS does not, and vice versa. Our main contribution is a simple set of two criteria for a large population ESS, which are similar (but not identical) to those originally proposed by Maynard Smith for infinite populations.     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.     

Evolutionary matrix games and optimization theory

An evolutionarily stable strategy (ESS) is only required to be capable of resisting invasion by rare mutant strategies. In contrast, an absolute invader strategy (AIS) is a rare mutant strategy that can invade any established strategy. We show that the predictions of the outcome of evolution made by optimization models are compatible with those made by the classical expected payoff comparisons in matrix games. We also show that if a matrix game has an AIS that AIS is unique and is also an ESS. But an ESS need not be an AIS. In pure-strategy submodels, an AIS need not be unique. An AIS of a matrix game has global asymptotic stability property in the game dynamics which involve only pure strategies including the AIS.     

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k>2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.     

欺骗,契约,合作的ESS

动物竞争中的欺骗、契约、合作现象是极为常见的。本文就论述与此有关的ESS。一、欺骗某些动物竞争时信号a后跟随发生行为A,如一个体不出现A则为欺骗。欺骗是相对真实     

Payoffs and strategies in territorial contests: ESS analyses of two ecotypes of the spiderAgelenopsis aperta

Summary Game-theoretic analyses were completed on the territorial contest behavior of two populations of a desert spider that exhibit markedly different levels of within-species competition. Numerical payoff matrices were constructed from field data collected on the behavior and demography of each population. Payoffs were expressed in terms of expected future egg production. Three behavior patterns that a spider might exhibit following assessment of its weight relative to that of its opponent and the value of the site were considered: withdraw, display, or escalate. The model predicts for the more harsh grassland habitat an evolutionarily stable strategy (ESS) that makes ownership decisive in settling contests between opponents with small weight differences, whereas it otherwise assigns victory to the heavier opponent. Whereas the empirical data collected for this grassland population closely approximates the predicted ESS, that for a population occupying a more favorable riparian habitat deviates significantly. The ESS prediction for this latter population is that an intruding spider will withdraw from a contest if it is similar in weight to the web-owner. Withdrawal is common in this population, but so are display and threat and these actions were not predicted. We hypothesize that gene flow from surrounding habitats is preventing the riparian population from completely adapting to its local environment.     

John Maynard Smith and the natural philosophy of␣adaptation

One of the most remarkable aspects of John Maynard Smith’s work was the fact that he devoted time both to doing science and to reflecting philosophically upon its methods and concepts. In this paper I offer a philosophical analysis of Maynard Smith’s approach to modelling phenotypic evolution in relation to three main themes. The first concerns the type of scientific understanding that ESS and optimality models give us. The second concerns the causal–historical aspect of stability analyses of adaptation. The third concerns the concept of evolutionary stability itself. Taken together, these three themes comprise what I call the natural philosophy of adaptation.     

Properties of a mixed ESS candidate in continuous strategy sets

An evolutionarily stable strategy (ESS) is a strategy that if almost all members of the population adopt, then this population cannot be invaded by any mutant strategy. An ESS is not necessarily a possible end point of the evolutionary process. Moreover, there are cases where the population evolves towards a strategy that is not an ESS. This paper studies the properties of a unique mixed ESS candidate in a continuous time animal conflict. A member of a group sized three finds itself at risk and needs the assistance of another group member to be saved. In this conflict, a player's strategy is to choose the probability distribution of the interval between the beginning of the game and the moment it assists the player which is at risk. We first assume that a player is only allowed to choose an exponential distribution, and show that in this case the ESS candidate is an attracting ESS; the population will always evolve towards this strategy, and once it is adopted by most members of the population it cannot be invaded by mutant strategies. Then, we extend the strategy sets and allow a player to choose any continuous distribution. We show that although this ESS candidate may no longer be an ESS, under fairly general conditions the population will tend towards it. This is done by characterizing types of strategies that if established in the population, can be invaded by this ESS candidate, and by presenting possible paths of transition from other types of common strategies to this ESS candidate.     

Alternative life histories in sex changing shrimp: a phenotype limited ESS

Summary In many populations of protandrous shrimp, two alternative life histories coexist. One way to interpret this fact is that one age (or size) group consists of a mixture of males and females. This is a nice example of a phenotype limited evolutionarily stable strategy (ESS), in the sense of Parker (1982, p. 187). This paper explores the ESS theory for the mixture.     

Long-term stability from fixation probabilities in finite populations: new perspectives for ESS theory

For mixed strategies in finite populations, long-term stability is defined with respect to the probability of fixation of a mutant. Under weak selection, necessary and sufficient conditions are obtained using a diffusion approximation of the Wright-Fisher model or exact solutions for the Moran model. These differ from the usual ESS conditions if the strategies affect fertility instead of viability, leading to a game matrix depending on the population size, or if the mutant mixed strategy uses a new pure strategy. In this case, the mutant deviation must not exceed some threshold value depending on the population size. In a diploid population, long-term stability may not occur unless there is partial dominance. In the case of sex allocation, continuous stability of an even sex ratio is ascertained. If sex allocation is random, an evolutionary decrease of the variance is predicted.     

Sex-limited mutations and the evolution of sexual dimorphism

Abstract.— Although the developmental and genetic mechanisms underlying sex differences are being elucidated in great detail in a number of species, there remains a breach between proximate and evolutionary studies of sexual dimorphism. More precisely, the evolution of sex-limited gene expression at autosomal loci has not been well reasoned using either theoretical or empirical methods. Here, I show that a Mendelian genetic model including elementary details of sexual differentiation provides novel insight into the evolution of sex differences via sex limitation. This model indicates that the nature of allelic effects and the pattern of selection must be known in both sexes to predict the evolution of sex differences. That is, selection interacts with genetic variation for sexual dimorphism to produce unanticipated patterns of trait divergence or convergence between the sexes. Ultimately, this model may explain why previous models for the evolution of sexual dimorphism do not predict the erratic behavior of the sex difference during artificial selection experiments.     

On evolutionarily stable sets

As an extension of the concept of an evolutionarily stable strategy (ESS) evolutionarily stable sets are introduced, i.e. sets of equilibrium strategies (EQS) which have much of the properties of an ESS. They are primarily used with evolutionary game models that allow a continuum of EQSs, none of which can be an ESS, but also include common ESSs as a special case. For a large class even of nonlinear models it can be shown that the standard dynamics converge towards some equilibrium point in an ES set if started within a neighbourhood of the set. Important applications of ES sets include e.g. mixed-strategist models and evolutionary game models in sexual populations.     

On sex allocation and selfing in higher plants

Summary Sex allocation (male allocation/female allocation) as a function of selfing rate is studied in the wild riceOryza perennis. Using dry weight measures, the male/female ratio is linearly related to the selfing rate. This linear relationship may have a fairly radical interpretation in terms of current sex allocation theory. It suggests that the intermediate selfing rates are themselves maintained by a form of frequency dependence. In particular, the linearity suggests: (i) the relative fitness of a selfed versus outcrossed offspring decreases with increased selfing; (ii) in equilibrium, a selfed offspring is approximately half as fit as an outcrossed offspring; (iii) the frequency dependence, being the opposite of that proposed in most selfing models, may result from the same forces thought to be involved in the maintenance of sex itself, and (iv) the position of the fitted line contains information about the plant's use of wind pollination for male reproduction. It suggests that wind shows much less mixing of pollen than previously assumed, and/or that there are severe morphological constraints on pollen presentation. The above interpretations are clearly speculative and tentative. Possible problems in the analysis, and some alternatives for data interpretation are discussed.     

Intergall migration in aphids; a model and a test of ESS dispersal rate

Summary I present an inclusive-fitness model for the evolution of dispersal rates of the offspring of asexual organisms living in discrete sites, which vary in available resources. I also assume a stable and saturated condition and that the offspring can respond to the variation in the capacity (amount of resources) of their natal sites. The model was tested using data obtained from the intergall migration in the yezo-spruce gall aphid,Adelges japonicus. All the parameters needed for the model, which included the cost of dispersal, both dispersal rates and available resources in each site, were estimated from field examinations. The data fit the model well, suggesting the importance of kin selection in determining the dispersal rates. Both actual and ESS dispersal rates are shown as concave functions of site capacity with a minimum rate for intermediate site capacity. The effect of both actual and ESS dispersal is to reduce, but not eliminate sibling competition within natal sites, which is most severe in intermediate site capacity.     

Effects of encounter in a population of spatial prisoner’s dilemma players

We study the evolution of cooperation in spatial prisoner’s dilemma games, whereby each player extends its interaction scope by trying to interact with a certain number of encounters randomly chosen from its non-neighbors, in addition to its permanently linked nearest neighbors. Furthermore, the non-neighbors treat the initiative interactions in two scenarios: definitely accepting that from the cooperators, whereas guardedly interacting with defectors with an acceptance probability which may take arbitrary value in [0,1]. Importantly, our results reveal that the proposed encounter mechanism is a potent extrinsic factor that is able to boost cooperation when appropriately adjusting the values of the encounter number and acceptance probability, though rational players would always defect in one-shot encounters, regardless of the action from the counterparts. We hope our studies may help understand that the proposed encounter mechanism is also an important ingredient of a flourishing cooperative society.     

On the stochastic model for estimation of mutational distance between homologous proteins

Summary A set of simple equations is derived which gives the relationship between the observed amino acid differences per 100 codons and the evolutionary distance per 100 codons using Holmquist's stochastic model of molecular evolution.Contribution No. 910 from the National Institute of Genetics, Mishima, Shizuoka-ken 411 Japan.